This paper is appearing in the The B.E. Journal of Theoretical Economics. Please cite this article as:
Ormos, M., Timotity, D., The Case of “Less is More”: Modelling Risk-Preference with Expected
Downside Risk, The B.E. Journal of Theoretical Economics (2017), DOI: 10.1515/bejte-2016-0100
This is the pre-print version of the accepted paper before typesetting.
The case of “Less is more”
Modelling risk-preference with Expected Downside Risk
MIHÁLY ORMOS
Department of Finance
Budapest University of Technology and Economics
Magyar tudosok krt 2., Budapest, Hungary
[email protected]
DUSÁN TIMOTITY
Department of Finance
Budapest University of Technology and Economics
Magyar tudosok krt 2., Budapest, Hungary
[email protected]
Abstract
This paper discusses an alternative explanation for the empirical findings contradicting
the positive relationship between risk (variance) and reward (expected return). We show
that these contradicting results might be due to the false definition of risk-perception,
which we correct by introducing Expected Downside Risk (EDR). The EDR parameter,
similar to the Expected Shortfall or Conditional Value-at-Risk, measures the tail risk,
however, fits and better explains the utility perception of investors. Our results indicate
that when using the EDR as risk measure, both the positive and negative relationship
between expected return and risk can be derived under standard conditions (e.g.
expected utility theory and positive risk-aversion). Therefore, no alternative
psychological explanation or additional boundary condition on utility theory is required to
explain the phenomenon. Furthermore, we show empirically that it is a more precise
linear predictor of expected return than volatility, both for individual assets and
portfolios.
Keywords: asset pricing; variance; conditional value at risk; expected downside risk;
utility theory; behavioral finance
JEL classification: G02; G12 G17; C53; C62
Acknowledgements: We would like to gratefully acknowledge the valuable comments
and suggestions of two anonymous referees that contribute to a substantially improved
paper. Mihály Ormos acknowledges that this study was supported by the János Bolyai
Research Scholarship of the Hungarian Academy of Sciences
1
1. Introduction
According to Modern Portfolio Theory (MPT) (Markowitz 1959), if returns follow elliptical
distributions, the utility of stochastic investment opportunities can be described as the
following, by assuming either wealth-dependent (constant relative, CRRA) or wealthindependent (constant absolute, CARA) risk-aversion under Expected Utility Theory
(EUT) (Gossen 1854). In either case, the Taylor approximation of expected utility yields
the following relationship between expected return and variance:
𝑈(𝐹) ≅ 𝐸(𝐹) − 0.5𝑎𝜎
(1)
where U(F), E(F) and σ stand for the utility, expected value and standard deviation of a
the possible realizations of a one-period investment, while a is an Arrow-Pratt measure
of absolute risk-aversion. By assuming unit wealth, this well-known equation in asset
pricing (e.g. Capital Asset Pricing Model by Lintner 1965; Mossin 1966; Sharpe 1964)
suggests a positive relationship between expected nominal changes (or if unit wealth is
assumed, expected returns) and volatility under standard circumstances (i.e. positive
risk-aversion coefficient) Therefore, regardless of the utility function used (CARA or
CRRA), equation (1) holds.
However, contradicting empirical results (Brooks et al. 2014; Kahneman and Tversky
1979, Tversky and Kahneman 1992; Linville and Fischer 1991; Post and Levy 2005)
indicate a negative relationship between the two parameters leading to the emergence
of an alternative utility theory: the prospect theory of Kahneman and Tversky
2
(Kahneman and Tversky 1979). The authors find that in certain cases (e.g. decision on
losses) if investors have two options yielding the same expected return they tend to
choose the option involving higher risk. They propose a convex, fluctuation dependent
utility function for losses as a solution to the problem.
In contrast, we argue that this behavior deviating from that suggested by the EUT is not
necessarily due to a flaw in the theory itself but the measurement of the risk perception.
Although numerous novel risk measures have emerged since the MPT, such as Valueat-Risk (VaR) (Campbell et al. 2001; Jorion 2007) or Conditional Value-at-Risk (CVaR)
(Rockafellar and Uryasev 2000; Acerbi and Tasche 2002) or entropy (Ormos and
Zibriczky 2014), none of them had the initial purpose of contributing to equilibrium
modelling by describing risk-preference in a more precise way. Therefore, we introduce
Expected Downside Risk (EDR) (Ormos and Timotity, 2016a, 2016b), based on CVaR,
with all its advantages but without its disadvantage of using a pre-defined probability
level. Our proposed measure is effectively the expectation of returns below the
expected return, or in other words, the expected bad outcome. Hence, in contrast to
standard risk-measures (e.g. volatility or VaR), higher EDR means lower risk, and
requires lower expected return as compensation for risk-aversion.
Subsequent to defining our proposed risk measure and its theoretical application in
asset pricing, we provide an empirical analysis on its explanatory power of the average
return of individual assets as well as well diversified portfolios, test whether the
theoretical findings hold in reality as well and make comparison with its peers such as
volatility, variance and 5% Conditional Value-at-Risk. As the main contribution of this
paper, we show that, without leaving the standard EUT framework, both the theoretically
3
optimal portfolio and the empirically best fitting linear model can reflect both negative
and positive risk-return relationship in our proposed model.
The rest of the paper is organized as follows: in Section 2 we introduce and discuss our
proposed method for measuring risk perception, in Section 3 the empirical investigation
of our proposed explanation is presented and lastly in Section 4 we provide a brief
conclusion.
2. The Model
We argue that the flaw in the conclusion of contradicting experimental and empirical
results on the relationship between risk and return comes from the fact that the tests try
to explain the results in a volatility or variance based setting. This setting, as suggested
by equation (1), indeed yields the result that investors having dominantly positive riskaversion coefficient (Barsky et al. 1997; Hanna and Lindamood 2004) should choose an
investment option involving a lower risk for a given level of expected return.
In contrast, in line with prospect theory, we argue that investors do not focus on (in
practical terms, they do not perceive) the volatility or variance of their investment but the
expected loss of it. However, we do not assume that investors should behave according
to Prospect Theory, that is, they do not necessarily follow loss aversion instead of risk
aversion. We argue that keeping the EUT setting with constant risk-aversion while
changing the applied risk measure, one may explain the contradicting results of riskand loss-aversion.
4
This change of risk measure also allows for including other properties of the return
distribution (e.g. skewness) that have been documented to play an important role in
explaining the risk premium (Astebro et al. 2014; Post et al. 2008). There are some
important risk measures that cope with asymmetric distributions, and still can be used
as coherent measures (Artzner et al. 1999), and are supported by general equilibrium
(Csoka et al., 2007), such as the Conditional Value-at-Risk, but are defined with an adhoc probability level that questions their robustness.
Our proposed measure is the expected loss of an investment labelled as the Expected
Downside Risk (EDR), which is defined in the following way: similar to the Conditional
Value-at-Risk, EDR measures the expected tail risk; however, it does not apply an adhoc probability level and considers returns below the expected return as a loss, that is
𝐸𝐷𝑅(𝑥) = 𝑝(𝑟(𝑥) ≤ 𝐸(𝑟(𝑥)))
∫
( )
( ( ))
𝑟 (𝑦)𝑝(𝑦)𝑑𝑦.
(2)
where 𝑟(𝑥) and 𝐸(𝑟(𝑥)) sign the return and its expectation of a given portfolio x and
𝑟 (𝑦) and 𝑝(𝑦) stand for the outcomes of the portfolio x and their respective
probabilities. In other words, EDR, as its name suggests, measures the expected loss
(risk) of investors given that their reference point is the expected return as suggested by
Easterlin (1974) or Kőszegi and Rabin (2006).
By looking at the aforementioned definition, one may find EDR very similar to Expected
Shortfall (ES) or Conditional Value-at-Risk (CVaR). In fact, depending on skewness of
the return distribution, EDR is equal to the CVaR calculated with the quantile of the
median with an opposite sign. For example, for symmetric distributions, where the
5
median and the mean are located at the same point, EDR is always defined as the
opposite of the 50% CVaR.
This change of risk measure is a plausible modification of standard asset pricing models
assuming EUT for two reasons. On the one hand, due to bounded rationality (Simon
1982), measures being easier to interpret are the main factors driving the focus of
investors (Gigerenzer and Selten 2002); in particular, this is an important reason why
fixed monetary payments and assigned probabilities are applied in experimental and
laboratory tests instead of complicated mathematical formulas (Tversky and Kahneman
1992), such as a variance-expected return choice set. Moreover, the precise prediction
of such measures for the future requires even more complex techniques (Andor and
Bohák, 2016; Ormos and Timotity, 2016c). Therefore, the expected amount of money
an investor can lose might be of greater relevance than the expectation of the squared
deviations from the mean. This idea is in line with recent studies finding that separately
applying losses and gains in asset pricing models yield higher goodness-of-fit (Tsai et
al., 2014; Cheng et al., 2014). On the other hand, the variance based approach yields
the optimal portfolio choice only if each compared investment has the exact
standardized distribution (i.e. excluding the variance and mean, they are identical). This
problem has already been solved by Value-at-Risk and Expected Shortfall due to their
non-parametric approach, however, both have a pre-defined probability-level that
cannot be verified fundamentally (i.e. the reason for the choice of a given probability
level does not have an economic explanation). Nonetheless, in our empirical results
section we compare EDR against these risk measures as well using the most popular
probability levels.
6
Assuming that, apart from the expected return and variance, the distribution type of the
return of a given asset does not change over time (which, in reality, is a fairly valid
assumption (Singleton and Wingender 1986; Sun and Yan 2003)) its EDR can be
described as a linear function of its expected return and volatility of past return. In order
to illustrate this relationship we provide an example with normal distribution as
approximation, which allows for tighter conditions; however real distributions of returns
can be calculated in the same way by changing the coefficient only. Below, we provide
our first theorem and its proof that derives the risk-return relationship in the asset pricing
model based on the EUT.
Theorem 1: Indifference curves are quadratic functions of the EDR and the expected
return of a distribution.
Proof: We can define EDR as the function of expected return and standard deviation:1
(3)
𝐸𝐷𝑅(𝑥) = 𝐸(𝑟(𝑥)) − 0.8𝜎
According to this equation, we can substitute the volatility (σ) with EDR in equation (1).
Therefore, the approximating function can be implemented in the EDR-E(r) setting:
𝑈 = 𝐸(𝑟(𝑥)) −
1in
.
.
𝑎[𝐸(𝑟(𝑥)) − 𝐸𝐷𝑅(𝑥)] .
the case of normal distribution EDR=-CVaR0.5= ∫
( )
[
𝑟∙
√
𝑒
( )]
(4) □
𝑑𝑟 that is equal to 0.8 assuming standard
normal distribution, and therefore, adding a constant (the expected return) and a multiplication by the standard
deviation yield equation (3)
7
In Figure 1 we provide a numerical simulation of the indifference curve for an average
constant absolute risk-aversion coefficient (Barsky et al. 1997) of a=4, where points
signed by +, - and x represent utility levels of U=1, 2 and 3 respectively. Here, we
underline that this generally accepted level of absolute risk-aversion is approximately
equal to the relative risk-aversion measure of investors behaving in a similar pattern;
therefore, again, one can use both utility functions in the equation above. Although
these examples use normal distribution with the 0.8 volatility coefficient seen in equation
(3), we show later that in general the coefficient is actually not far from this value.
Please insert Figure 1 here
Furthermore, the generalized form of equation (4) yields
0 = −𝑐
𝐸(𝑟(𝑥)) −
( )
+ 𝐸𝐷𝑅(𝑟(𝑥)) +
− 𝑈,
(5)
where U stands for the utility and c is a constant for each investor defined as
𝑐=
.
.
𝑎.
(6)
Equation (5) clearly indicates the quadratic indifference curve, however, it also suggests
a more intuitive result: since the first term is always non-positive (assuming a positive
risk-aversion coefficient) the following equation must hold in order to have a real
solution, that is
8
𝐸𝐷𝑅(𝑥) +
(7)
− 𝑈 ≥ 0.
For further analysis of our asset pricing model, we define the slope of the indifference
curve in the EDR-E(r) setting. In the following theorem, we highlight the main theoretical
contribution of our paper.
Theorem 2: In the EDR-E(r) system, both a negative and a positive relationship
between risk and return can be derived under standard, EUT conditions.
Proof: In the followings, we use the general, parametric approach, hence, the deduction
is valid for any distribution. In this generalization, instead of the 0.8 level, we assign to v
a distribution-dependent coefficient of the volatility, that is 𝑣 =
( ( ))
( )
. We know
that the total derivative of the indifference curve should be zero; therefore, the slope can
be calculated as in equation (8).
( )
=
𝐸(𝑟(𝑥)) − 𝐸𝐷𝑅(𝑥) +
( ( ))
( )
1−
𝐸(𝑟(𝑥)) − 𝐸𝐷𝑅(𝑥)
= 0.
(8)
From equation (8), the sensitivity of the expected return for Expected Downside Risk
can be expressed as
( ( ))
( )
=−
( ( ))
( ( ))
( )
( )
=1−
9
( ( ))
( )
.
(9)
This relationship implies that if 𝐸𝐷𝑅(𝑥) < 𝐸(𝑟(𝑥)) −
, the slope of the indifference
curve is positive and greater than one
( )
( )
if 𝐸𝐷𝑅(𝑥) = 𝐸(𝑟(𝑥)) −
𝐸(𝑟(𝑥)) −
> 1,
(10)
, the slope is positive or negative infinity, and if 𝐸𝐷𝑅(𝑥) >
, the slope can be both negative and positive, but less than one
( )
( )
< 1.
(11)
Furthermore, the expectation of returns not greater than the expected return cannot be
higher than the expected return itself; hence, including the constraint of 𝐸𝐷𝑅(𝑥) ≤
𝐸(𝑟(𝑥)),
𝐸(𝑟(𝑥)) −
< 𝐸𝐷𝑅(𝑥) ≤ 𝐸(𝑟(𝑥))
(12)
yields the “usual” positive relationship between risk and expected return. However, in
equation (10) we have shown that in the case of small EDRs the relationship changes
and a higher risk will be rewarded with lower expected return.
10
□
The aforementioned relationship is presented in Figure 2, where the curve, the dashed
line and the dotted line stand for the indifference curve, the risk-free portfolios and the
𝐸𝐷𝑅(𝑥) = 𝐸(𝑟(𝑥)) −
constraint respectively. Here we use a=0.5 and v=0.8
parameters.
Please insert Figure 2 here
Since the appearance of the CAPM, asset pricing models also implement the effects of
leverage opportunities at the risk-free rate. Therefore, in the followings, we also show
that the aforementioned, changing relationship between risk and reward shows up in the
leveraged portfolio optimization as well.
Theorem 3: If leverage is included in the EDR-E(r) setting, portfolio optimization could
yield both negative and positive slopes for the “capital market line” collecting the
efficient portfolios.
Proof: In order to analyze the case including leverage opportunity, first we have to
define whether it provides an optimal solution as well. Here, we assume a unique riskfree asset that provides risk-free return 𝑟 . In line with the standard volatility – expected
return-based Markowitz model or the Beta-based CAPM, the weight of risk-free asset in
the portfolio affects linearly both the EDR and the E(r). Therefore, the expected return of
the leveraged portfolio can be defined as a linear function of EDR, where the ratio
between risk and return or the price of risk (𝑘 in equation (13)) is constant, which is
similar to the security market line of the CAPM. This finding further implies that, if an
optimal solution exists, it is equal to the point of tangent of the leveraged portfolios of
11
𝐸(𝑟(𝑥)) = 𝑟 + 𝑘𝐸𝐷𝑅(𝑥),
(13)
and the highest indifference curve
𝑈 = 𝐸(𝑟(𝑥)) −
.
𝑎[𝐸(𝑟(𝑥)) − 𝐸𝐷𝑅(𝑥)] .
(14)
Based on investors’ goal of maximizing their utility we get a simple linear constraint
optimization problem from equation (13) and (14), which is
max
( ),
( )
𝐸(𝑟(𝑥)) −
.
𝑎[𝐸(𝑟(𝑥)) − 𝐸𝐷𝑅(𝑥)]
𝑠. 𝑡. 𝐸(𝑟(𝑥)) = 𝑟 + 𝑘𝐸𝐷𝑅(𝑥) .
(15)
Using the Lagrangian and its derivatives we get
𝐿 = 𝐸(𝑟(𝑥)) −
.
𝑎[𝐸(𝑟(𝑥)) − 𝐸𝐷𝑅(𝑥)] + 𝜆 𝐸(𝑟(𝑥)) − 𝑟 − 𝑘𝐸𝐷𝑅(𝑥) ,
(16)
[𝐸(𝑟(𝑥)) − 𝐸𝐷𝑅(𝑥)] + 𝜆 = 0,
(17)
( ( ))
( )
= 1−
=
[𝐸(𝑟(𝑥)) − 𝐸𝐷𝑅(𝑥)] − 𝜆𝑘 = 0,
= 𝐸(𝑟(𝑥)) − 𝑟 − 𝑘𝐸𝐷𝑅(𝑥) = 0.
12
(18)
(19)
According to eq. (17)-(18) the optimal solution for the constrained optimization problem
is
𝐸𝐷𝑅(𝑥)
=
𝐸(𝑟(𝑥))
=
(
)
(
)
We further have the condition of 𝐸(𝑟(𝑥))
−
+
(20)
,
(
−
)
(21)
.
, therefore, the solution is valid
≥ 𝐸𝐷𝑅(𝑥)
if and only if
(
)
(22)
≥0
(23)
𝑘 ∈ (1, ∞) ∪ (−∞, 0].
In order to define the optimal portfolio that is used in leveraging we determine the slope
coefficient in the following way. Substituting back into eq. (14) we get
𝑈=
(
)
+
(
)
−
−
.
𝑎
(
)
Then the derivative of (22) is
13
=
.
(
)
+
(
)
.
(24)
(
=
)
(
=
)
(
(25)
= 0,
)
we find that
=
(
)
(
(
)
)
(
=
)
(
,
)
(26)
which is always positive under the first order condition (25), therefore
=
if
= 0 where 𝑘 =
(
)
(27)
>0
stands for the extremum of the slope coefficient. These
together yield that the utility function does not have a local maximum but a minimum (in
line with Figure 1 and Figure 2). Furthermore, as the derivative of the leveraged utility
function with respect to the slope coefficient depends on both the slope and the given
risk-aversion, risk-free return and volatility combination, we define the following cases:
If 𝑎𝑟 > 𝑣
⎧> 0 𝑖𝑓
⎪< 0 𝑖𝑓
⎨
⎪> 0 𝑖𝑓
⎩
and if 𝑎𝑟 < 𝑣
14
𝑘<0
1<𝑘<
𝑘>
(26)
,
⎧< 0 𝑖𝑓
⎪
> 0 𝑖𝑓
⎨
⎪< 0 𝑖𝑓
⎩
𝑘<
(27)
<𝑘<0
𝑘 > 1.
These are the main theoretical findings of the proposed model. They indicate that
depending on the exogenous parameters of 𝑎, 𝑟 , 𝑣 (i.e. the risk-aversion coefficient,
the risk-free rate and the coefficient of volatility in the EDR regression) the leveraged
optimization could lead to both negative and positive slope choice. Investors maximize
utility either by choosing the leverage slope closest to the EDR=E(r) line or by they
picking the leveraged portfolio line the furthest from the 45 degree line.
The theoretical maximum of utility would be reached at the leveraged line with slope
closest to unity since the limit from the right (getting closer to the EDR-E(r) line on the
positive side)
.
lim 𝑈 = lim
→
(
→
+
)
(
)
=∞
(28)
is infinite, while the leverage portfolio line the furthest yield only a finite utility since
lim 𝑈 = lim
→
→
.
(
)
+
(
)
=
.
(29)
Therefore, investors would theoretically prefer the leveraged line with slope of unity over
the one with the slope of zero, however, in the real world these portfolios are not always
15
attainable for investors. In these cases, the leveraged portfolio line with the highest
negative slope coefficient might be the optimal choice.
□
Figure 3 summarizes the aforementioned equations where line 1 and line 2 stand for the
𝐸𝐷𝑅 ≤ 𝐸(𝑟) constraint and the 𝑘 =
condition (i.e. the slope where utility is
minimal) respectively.
Please insert Figure 3 here
3. Empirical results
3.1 Data
In our empirical investigations we test the realization of the EDR-E(r) equilibrium for
equity portfolios, moreover, we analyze how the equilibrium is parameterized; first, for
unleveraged portfolios, and second, for leveraged ones. We apply daily and annual data
and statistics to investigate unlevered and levered portfolios. The data used for these
portfolio calculations consist of daily returns from July 31, 1993 to July 31, 2014 of the
340 constituents of S&P 500 index that have been listed both at the beginning and at
the end of the period. Thus the dataset we use is not free of survivorship bias. In order
to model leveraged portfolios we apply the mean annualized 3-month T-bill log return for
the same 21 years. This dataset is used in sections 3.2 and 3.4.
In section 3.3, where no individual asset is considered, we apply longer historical time
series: we use the annual returns of the S&P 500 index and the 3-month Treasury bills
between 1928 and 2013.
16
3.2 Optimization for unleveraged and leveraged portfolios
In the case of unleveraged portfolios we simulate the performance of 10,000 randomly
weighted portfolios consisting of the 340 different stocks. The EDR calculation is based
on either daily or yearly returns. The plot of these simulated portfolios using yearly
statistics is presented in Figure 4. One can see that the concave efficient frontier
representing portfolios with the highest EDR/E(r) ratio always has an optimum at the
point of the tangent with the convex utility functions (as shown in Figure 1).
Please insert Figure 4 here
Our utility maximization results based on the simulated EDR-E(r) pairs indicate that for
a=0.5 (extremely low) and a=4 (average) risk-aversion the {EDR,E(r)}={0.15, 0.27}
portfolio is optimal on the simulated set of possible portfolios; however, for a=10
(extremely high) the {0.15, 0.24} is optimal. Here, we find evidence of the surprising
utility preference we have derived in the previous section: for a given level of risk
(measured by the EDR), a portfolio with a lower expected return may provide higher
utility; in particular, the utilities generated by equation (14) are shown in Table 1. The
results show two important patterns: on the one hand, for fixed expected return the
increasing risk-aversion, as expected, decreases the utility of the stochastic payoff; on
the other hand, for fixed risk-aversion, one may clearly see that for a=0.5 and a=4 a
decrease in the expected return yields a loss of utility, which is in line with standard
asset pricing theories, however, this relationship is the opposite for a=10.
17
Please insert Table 1 here
This latter finding is due to the following mechanism: since EDR is fixed here, a
decrease in the expected return comes with a proportional drop in return volatility;
initially, at low risk-aversion levels (a=0.5, a=4), the drop in the expected return hurts
more than the excess utility provided by the decreasing volatility; however, at very high
risk-aversion levels, the latter becomes larger in magnitude and can counterbalance the
loss of expected return. Nevertheless, using the EDR as risk-measure, one can see
both increasing (a=0.5, a=4) and decreasing utility (a=10) as a function of the expected
return.
We also run the simulation using daily statistics in Figure 5. Using a similar dataset (as
above) at a daily level, we find that the optimal portfolios are much further away from
each other than in the annual analysis. This is mainly due to the fact that the point of
tangent between the efficient frontier and the indifference curve is on the decreasing
part of the latter.
Please insert Figure 5 here
In case of leveraged portfolios we first apply the slope optimization described in Section
2. In order to illustrate this optimization let us include a risk-free rate in the
aforementioned portfolio simulation. Using the mean annualized 3-month T-bill log
return for the same 21 years we get a risk-free rate (𝑟 ) of 2.73%. In addition, applying
the OLS estimation of
18
(28)
𝐸(𝑟(𝑥)) − 𝐸𝐷𝑅(𝑥) = 𝑣𝜎
to the yearly portfolio simulation results yields a fitted volatility coefficient of v=0.78. The
difference from normality (i.e. the theoretical volatility coefficient of 0.8) is well-reflected
here. For comparison purposes we mention that we measure daily, and monthly
volatility coefficients to be 0.68 and 0.76 respectively, which are in line with the standard
findings that deviation from normality (e.g. fat tails) decreases as the investment horizon
increases.
These parameters further lead to the definition of 𝑘 and yield the sign of the EDR-E(r)
relationship in the following way: according to the simulated portfolios, the leveraged
portfolio lines have the slope of
( ( ))
( )
= 1.64 and
( ( ))
( )
= 0 at the boundaries (the
closest to and the furthest from the EDR=E(r) line). Utility at the boundaries is obtained
by substituting back into eq. (24). Analysis of the utility difference of 𝑈
yields that for ∈ (0,28.55) 𝑈
.
>𝑈
.
−𝑈
, hence, a positive relationship between EDR
and E(r) is preferred. Figure 6 shows the leveraged portfolio optimization conditional on
yearly parameters, where the red “+” sign stands for the risk-free asset while the
horizontal and
( ( ))
( )
= 1.64 sloped lines represent the optimal leverage line given 𝑎 ≥
28.55 and 𝑎 < 28.55 respectively. According to this optimization investors having 𝑎 <
28.55 (the majority according to Barsky et al. (1997) and Hanna and Lindamood (2004))
choose from the leveraged portfolios on the EDR=rf+1.64E(r) line, while those having
𝑎 ≥ 28.55 invest into portfolios providing EDR=rf.
19
Please insert Figure 6 here
The daily analysis yields somewhat different results. Here, we measure an average 𝑟 =
0.011%, v=0.68 and the boundary slopes range from 0 to -0.08. This case implies that
for ∈ (0,155.69) 𝑈
.
>𝑈
. This simulation is shown in Figure 7. It means that
practically every investor picks portfolios from the EDR=rf-0.08E(r) line, and therefore,
we see a negative relationship between EDR and E(r) at the daily frequency. This
negative relationship between risk and reward in the short term is well in line with recent
literature, which confirms the existence of investors increasing portfolio risk immediately
subsequent to negative asset price shocks (Ormos and Timotity, 2016d; Ormos and
Timotity, 2016e).
Please insert Figure 7 here
3.3 Ambiguous risk-preferences
Here we illustrate the aforementioned optimization with the following simple portfolio
choice problem. We measure the following parameters of a well-diversified portfolio (we
assume the S&P500 index behaving as the market portfolio): the average log return
𝐸(𝑟 ) = 9.12%, the annual return volatility 𝜎 = 19.5%, 𝐸𝐷𝑅 = −13.07% and 𝑟 =
3.47%. The v parameter introduced in equation (8) hence takes on 𝑣 = 1.14. Based on
Barsky et al. (1997) and Hanna and Lindamood (2004) we assume an average investor
having a risk-aversion coefficient of 𝑎 = 4.
20
Now let us look at the boundary portfolio providing minimal EDR where equation (9)
yields an infinite indifference curve slope. We know that equation (3) and its
modification with the volatility coefficient still holds, therefore
𝐸𝐷𝑅 = 𝐸(𝑟) −
= 𝐸(𝑟) − 𝑣𝜎 .
(29)
Then, equation (30) follows as
𝜎 = .
(30)
We further know that leveraged positions combined with the risk-free interest rate yield
the following equation
𝐸(𝑟) = 𝑟 +
(
)
𝜎 .
(31)
The solution of the equation system yields 𝐸(𝑟) = 11.72%, 𝜎 = 0.2846, 𝐸𝐷𝑅 =
−20.67% and 𝑈 = −0.0447. Now let us measure the indifference curve of the same
investor by testing the EDR-E(r) pairs with “decreased” risk. Here we mean decreased
risk by increasing EDR for a given utility level, that is, let us consider a portfolio X where
𝐸𝐷𝑅 = −10%. By the definition of the indifference curve 𝑈 = 𝑈 , which gives two
possible solutions of 𝐸(𝑟) =
−3.89%
0.0472
with 𝜎 =
. In other words, given the
48.68%
0.4529
21
type of the return distribution constant and the expected loss fixed at -10%, investors
having a risk-aversion coefficient of a=4 are indifferent to the choice between a portfolio
providing an expected return of -4% or 49%. This phenomenon might be surprising if
variance is not taken into consideration (as in most of the experimental studies) and it
underlines the importance of measuring risk-preference.
3.4 Robustness test and the time component
Furthermore, we provide a robustness test by running a linear regression model for
annual, monthly and daily parameters as well. Here, we run regressions at daily,
monthly and yearly return horizon for daily overlapping periods. The applied
methodology consists of univariate OLS regressions, in which the predictor variables
are volatility, 5% Value-at-Risk, 5% Conditional Value-at-Risk, CAPM Beta and EDR
respectively. As noted before, the filtered sample for individual shares consists of 340
assets (i.e. 340 expected returns, and measures of risk accordingly), whereas for
portfolios we use 10,000 randomized, long-only portfolios. The purpose of these
regressions is to test whether EDR is important enough for investors to bear a risk
premium, and whether it performs better against its competitors from standard
univariate asset pricing models. In Table 2 we present the results of the simple linear
model’s estimating expected return.
Please insert Table 2 here
In contrast to the estimated positive relationship between return and risk of the volatilitybased model in any terms, changing the risk measure to EDR reveals an interesting
22
pattern: expected return and risk are only positively correlated in the short run.
Regression on annual and monthly statistics indicate a positive relationship between
EDR and E(r) both for individual assets and portfolios. This means that as the
expectation of the negative outcomes increases (the risk decreases), the expected
return increases as well, which is against the findings of standard asset pricing theories.
Furthermore, analyzing the p-values reveals that pricing regressions indicate that
investors indeed seem to focus more on the expected loss instead of the simple risk
measures as volatility and variance. Moreover, as the p-value is lower in EDR
regressions than in VaR and CVaR models, it seems that one considers the losses on
the whole domain instead of taking into account the tail risk only (e.g. at the 5%
probability level analyzed). EDR surpasses in goodness-of-fit the widely used CAPM
and its Beta measure as well in five out of the six tests, however, the Beta yields much
worse, insignificant estimations for individual assets. Altogether, we conclude that EDR
seems to be a more precise estimator of E(r) than its competitors.
In order to analyze the effect of the period length on the leveraged portfolio line, we run
linear regressions for distinct intervals and test whether the coefficients between risk
and return show a robust pattern. Figure 8 represents our results indicating a robust
positive relationship between the coefficient of EDR and the length of the analyzed
period. Again, this phenomenon implies that focusing on the very short term, investors
seem to be risk-averse and at around 11-12 days they are insensitive to risk; for periods
over 12 days a negative relationship between risk and return applies to them. Here, we
highlight again that this risk-seeking behavior exists in the sense that portfolios with
different expected return may provide the same utility for given expected downside risk;
23
however, it does not reject the Expected Utility Theory or require negative risk-aversion
coefficients.
Please insert Figure 8 here
4. Concluding remarks
Numerous studies found evidence of investors’ behavior contradicting the well-known
positive risk-reward relationship. However, asset pricing theories based on the expected
utility theory have not yet given any explanation of a negative relationship under
standard circumstances. Experimental evidences show that the risk-aversion coefficient
is positive for a dominant portion of investors (and hence they are risk-averse),
although, in some cases they systematically behave in a seemingly risk-seeking way.
In this paper we argue that this deviation from the expected behavior might be due to
the false definition of risk-perception instead of a flaw in the definition of the perceived
utility (as previous studies suggest). Therefore, in our model we propose the Expected
Downside Risk as an alternative risk measure that describes and better fits investors’
sensitivity to risk as measured by pricing regressions. Being a simple measure of the
average loss relative to the expected return EDR seems to yield a more perceived value
of risk than the standard deviation from the historical mean, especially in the presence
of a highly discrete number of choices (e.g. experimental tests). This latter statement is
confirmed by our regression results as well, where expected return is more significantly
driven by EDR than volatility in all three period lengths for both individual assets and
portfolios.
24
Having confirmed that EDR measures risk-perception better than volatility; asset pricing
models based on expected utility theory can be modified by replacing the latter with the
former risk measure. The solution of the optimization problem reveals that both a
positive and negative relationship between expected return and EDR may exist under
standard circumstances.
Finally, our model is supported by regression results indicating a negative relationship
between risk and expected return for periods over 12 days and the “usual” positive
relationship for less than 12 days.
For further research one may include the test of Expected Downside Risk in
experimental settings, the detailed analysis of the effect of time on the risk-return
relationship measured by EDR or by examining portfolio optimization where both
negative and positive relationship can be found depending on the exogenous
parameters, the latter of which could yield an alternative method for measuring riskaversion.
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Table 1: Utilities at different risk-aversion for fixed EDR=0.15
E(U|(E(r)=0.27)
E(U|E(r)=0.24)
a=0.5
0.2641
0.2367
a=4
0.2224
0.2132
a=10
0.1510
0.1731
Notes: Table 1 stands for the expected utilities of the pricing equation 𝑼 = 𝑬(𝒓(𝒙)) −
𝟎.𝟓
𝒗𝟐
𝒂[𝑬(𝒓(𝒙)) −
𝟐
𝑬𝑫𝑹(𝒙)] given EDR=0.15, E(r)=0.27 and E(r)=0.24 for a=0.5, a=4 and a=10 risk-aversion
coefficients.
Table 2: Linear estimations of the expected return
Individual shares
Yearly
Monthly
Daily
σ
VaR(5%)
CVaR(5%)
Portfolios
Beta
EDR
σ
VaR(5%)
CVaR(5%)
Beta
EDR
Coeff
2.7E-02
3.5E-02
2.0E-02
3.3E-04
1.6E-01
4.7E-02
6.5E-02
2.3E-02
7.9E-03
2.0E-01
p-value
1.4E-01
1.2E-01
1.7E-03
9.3E-01
8.3E-15
2.5E-35
6.7E-28
2.2E-55
8.2E-25
0.0E+00
Coeff
7.2E-03
2.7E-02
3.2E-03
-2.5E-04
1.2E-02
7.3E-03
2.3E-02
3.9E-03
1.1E-03
2.1E-02
p-value
2.9E-01
3.6E-01
2.3E-01
6.2E-01
2.2E-01
9.1E-08
1.3E-03
1.6E-12
4.6E-26
4.8E-28
Coeff
3.0E-03
6.2E-02
-1.1E-03
4.3E-06
-6.1E-03
2.1E-03
4.1E-02
-4.2E-04
8.6E-05
-4.6E-03
p-value
4.9E-02
4.0E-02
1.2E-01
9.0E-01
1.5E-02
1.0E-11
3.4E-08
2.7E-03
3.3E-35
3.0E-20
Notes: Table 2 represents the OLS estimations of the 𝑬(𝒓) = 𝜶 + 𝜷 ∙ 𝒚 where y stands for volatility
(σ), 5% Value-at-Risk (VaR 5%), 5% Conditional Value-at-risk (CvaR 5%), CAPM Beta and EDR
measures. The 𝜷 coefficients and their significance levels (p-values) are shown for individual
shares and randomly generated portfolios using yearly, monthly and daily returns.
30
Figure 1: Iso-utility functions in EDR-E(r) system given a=4
Notes: Figure 1 represents iso-utility functions in the EDR-E(r) system given a=4 risk-aversion
coefficient. The points signed by +, - and x represent utility levels of U=1, 2 and 3 respectively.
Figure 2: Limits of iso-utility functions in the EDR-E(r) system
Notes: Figure 2 represents the indifference curve (solid line), risk-free portfolios (dashed line), and
the 𝑬𝑫𝑹(𝒙) = 𝑬(𝒓(𝒙)) −
𝒗𝟐
𝒂
constraint (dotted line) respectively, given a=0.5 risk-aversion
coefficient and v=0.8 volatility coefficient.
31
Figure 3 here: Leveraged portfolio optimization
Notes: Figure 3 represents the leveraged portfolio optimization where line 1 and line 2 stand for
the 𝑬𝑫𝑹 ≤ 𝑬(𝒓) constraint and the 𝒌 =
𝒂𝒓𝒇
𝒂𝒓𝒇 𝒗𝟐
condition (i.e. the slope where utility is minimal)
respectively. Investors try to reach the leveraged portfolio line the furthest from line 2.
Figure 4: Unleveraged portfolios in EDR-E(r) system
Notes: Figure 4 represents the EDR-E(r) pairs of 10,000 randomly simulated portfolios of the 340
S&P500 members existent both at the beginning and at the end of the analyzed period. The gray
points stand for the optimal portfolio (i.e. providing the highest expected utility) given a=0.5 or a=4
and a=10 risk-aversion coefficients. The parameters are calculated using non-overlapping yearly
returns.
32
Figure 5: Unleveraged portfolios in daily analysis
Notes: Figure 4 represents the EDR-E(r) pairs of 10,000 randomly simulated portfolios of the 340
S&P500 members existent both at the beginning and at the end of the analyzed period. The gray
points stand for the optimal portfolio (i.e. providing the highest expected utility) given a=0.5, a=4
and a=4 risk-aversion coefficients respectively. The parameters are calculated using nonoverlapping daily returns.
Figure 6: Leveraged portfolios in EDR-E(r) system
Notes: Figure 6 shows the leveraged portfolio optimization conditional on yearly parameters,
where the “+” sign in the intersection of the lines stands for the risk-free asset, while the
horizontal and
𝒅𝑬(𝒓(𝒙))
𝒅𝑬𝑫𝑹(𝒙)
= 𝟏. 𝟔𝟒 sloped lines represent the optimal leveraged portfolio line given 𝒂 ≥
𝟐𝟖. 𝟓𝟓 and 𝒂 < 𝟐𝟖. 𝟓𝟓 respectively.
33
Figure 7: Leveraged portfolios in daily analysis
Notes: Figure 7 shows the leveraged portfolio optimization conditional on daily parameters, where
the “+” sign in the intersection of the lines stands for the risk-free asset while the horizontal and
𝒅𝑬(𝒓(𝒙))
𝒅𝑬𝑫𝑹(𝒙)
= −𝟎. 𝟎𝟖 sloped lines represent the optimal leveraged portfolio line given 𝒂 ≥ 𝟏𝟓𝟓. 𝟔𝟗 and
𝒂 < 𝟏𝟓𝟓. 𝟔𝟗 respectively.
Figure 8: Relationship between time, risk and return
Notes: Figure 8 represents horizon-dependent relationship between the β coefficient of EDR in the
𝑬(𝒓) = 𝜶 + 𝜷 ∙ 𝑬𝑫𝑹. This phenomenon implies that focusing on the very short term, investors seem
to be risk-averse, and therefore, their coefficient is positive. However, at around 11-12 days they
become insensitive to risk and for investment periods over 12 days a negative relationship
between risk and return applies to them. The dashed line stands for the zero value.
34